Assessing power grid reliability using rare event simulation · Any power ﬂow model starts with the deﬁnition of its topology. The topology can be modeled by a connected graph

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Assessing power grid reliability using rare event simulation

Wadman, W.S.

Publication date2015Document VersionFinal published version

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Citation for published version (APA):Wadman, W. S. (2015). Assessing power grid reliability using rare event simulation.

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Part II

L I T E R AT U R E I N V E S T I G AT I O N

2D E T E R M I N I S T I C P O W E R F L O W A N A LY S I S

Any power flow model starts with the definition of its topology. Thetopology can be modeled by a connected graph with N nodes (also calledbuses) and M edges (also called branches or connections). The connectionfrom node i to j is referred to as (i, j). In this chapter the power injections(both consumption and generation) are constant and deterministic. Wewill distinguish two models for the power flow equations: AlternatingCurrent (AC) and Direct Current (DC). Both AC and DC equations aresteady state equations: they relate the power injections at all nodes to thepower flows through all grid connections assuming the latter immediatelyreach an equilibrium state. As opposed to steady state power flow models,time-dependent power flow models exist too [10, 78], but we will notconsider them in this work. Such models employ time frames rangingfrom milliseconds to seconds and are often used to decide on suddencontingencies like a short circuit or lightning strike. Since a step sizeon the order of minutes will capture the typical variability of powerinjections, the steady state power flow equations are sufficiently accuratefor our purposes. As we will see, the DC power flow equations are lessaccurate than AC partly because nodal voltages are assumed to be equalover all nodes, but their linear form makes it an appealing model for fastcomputation and analytic tractability. The AC power flow equations arenonlinear and require numerical methods to compute the systems state.Many books on power flow analysis describe both AC and DC equationsin detail [18, 37, 39].

2.1 the alternating current power flow model

To avoid confusion with vector indices, ı denotes the imaginary unit. Wefirst introduce the main variables.

13

14 deterministic power flow analysis

• The complex nodal voltage Vi ∈ C at grid node i. We introduce thepolar notation

Vi := |Vi|eıδi , (2.1)

where |Vi| ∈ [0, ∞) is the voltage magnitude and δi ∈ (−π, π]is the voltage angle (or phase angle) at node i. The voltage dropover connection (i, j) is Vj − Vi, and the voltage on the ground isassumed to be zero.

• The complex current injection Ii ∈ C at node i.

• The complex current Iij ∈ CN×N flowing through connection (i, j)from node i to node j.

• The complex power Si ∈ C injected at node i. We can write

Si := Pi + ıQi, (2.2)

where the real part Pi ∈ R is called the active power and theimaginary part Qi ∈ R is called the reactive power. For both Pi andQi positive values correspond to net generation at node i, whereasnegative values correspond to net consumption at node i.

• The admittance yij ∈ C of connection (i, j), given by

yij :=1

Rij + ıXij,

with Rij ∈ R the resistance and Xij ∈ R the reactance of connection(i, j). There exists no connection between node i and j 6= i if andonly if yij = 0.

We will derive the AC power flow equations from the complex, vectorvalued generalizations of Kirchoff’s current law, Ohm’s law and thedefinition of power. Kirchoff’s current law states that at any node thesum of currents flowing into that node is equal to the sum of currentsflowing out of that node:

−Ii +N

∑k=1

Iik = 0, (2.3)

2.1 the alternating current power flow model 15

at each node i. Figure 2.1 shows an example.

1

2 3 4 5

6

I43 I45

I46

Figure 2.1: The current entering any junction is equal to the current leaving thatjunction: I4 = I43 + I46 + I45, with I4 the current injection at node 4.

Note that Kirchoff’s voltage law is also present but redundant in themodel: the directed sum of the voltages around any closed circuit will bydefinition of the nodal voltages always be equal to zero. The second lawof importance is Ohm’s law: the current through a connection betweentwo nodes is directly proportional to the voltage drop over these nodes.Ohm’s law can be stated by

yij(Vi −Vj) = Iij. (2.4)

Combining Kirchoff’s current law and Ohm’s law, we can write

Ii =N

∑j=1

YijVj, (2.5)

or

I = YV (2.6)

in matrix-vector notation for an appropriately chosen matrix Y ∈ CN×N .This matrix is called the admittance matrix. It is easy to check that

Yij :=

{−yij if i 6= j,∑Nk=1 yik if i = j,

(2.7)

are the elements of the admittance matrix. Note that for i 6= j, Yij = 0 ifand only if there is no connection between node i and j. In this sense,


Y encodes the topology of the power grid. Furthermore, yii denotesthe shunt or admittance-to-ground at node i, which can be used toensure that the nodal reactive power remains in a specified interval.Capacitor shunts and inductor shunts respectively inject or consumereactive power, resulting in a higher or lower nodal voltage, respectively.The shunt yii will cause an extra amount of current Iis to be injected atnode i. Since the voltage on the ground is zero by definition, we haveIis = yii(Vi − 0) = yiiVi. This means that in the admittance matrix, anextra term yii has to be added to the diagonal term Yii.

We conclude the discussion on the admittance matrix by introducingtwo different notations for Yij:

Yij = Gij + ıBij = |Yij|eıθij . (2.8)Here Gij, Bij ∈ R are the conductance and susceptance, respectively, ofconnection (i, j) if i 6= j. The most right-hand side expression is simply thepolar notation using admittance angle θij ∈ (−π, π]. Now we introducethe definition of complex power

Si = Vi I∗i ,

at node i, where x∗ denotes the complex conjugate of x. Then, using (2.2)and (2.5), the complex conjugate of Si is

S∗i = Pi − ıQi = V∗iN

∑j=1

YijVj. (2.9)

Using (2.1) and (2.8) we rewrite the right-hand side in polar notation

Pi − ıQi =N

∑j=1|ViYijVj|eı(θij+δj−δi).

Expanding this equation and equating real and reactive (i.e. imaginary)parts results in the AC Power Flow Equations (AC PFEs). That is,

Pi = −N

∑j=1|ViYijVj| cos(θij + δj − δi) for all nodes i, (2.10)

Qi = −N

∑j=1|ViYijVj| sin(θij + δj − δi) for all nodes i. (2.11)


The power flow equations (2.10) and (2.11) form the central model ofmany power flow analyses. The admittance matrix elements |Yij| and θijare given, as well as power injections Pi, Qi. The system of equationsshould be solved for the voltage angles δi at all nodes and the voltagemagnitudes |Vi| at most nodes, as will be explained in the next section.

2.1.1 A modified Newton-Raphson solver for the AC power flow equations

In this section we will describe the details of solving the AC PFEs (2.10) –(2.11) using a modified Newton-Raphson method. First, we distinguishthree types of grid nodes:

1. The slack node.A power flow model will contain exactly one slack node (also calledthe swing bus), where the residual power of the network is eithergenerated or consumed. Hence, no power flow equations have tobe solved at the slack node. In this thesis, node 1 is always theslack node. Its voltage magnitude |V1| is given and without loss ofgenerality, we set δ1 = 0.

2. PQ nodes.If node i is a PQ node, a specified amount of real power Pi andreactive power Qi is injected at that node. Voltage magnitude |Vi|and voltage angle δi are unknown in the AC PFEs for each PQnode i. Typical examples of PQ nodes are nodes where poweris consumed only, and thus they are also known as load nodes.However, small-scale generators often control the real and reactivepower, and we assume in this thesis that all nodes with an uncertainpower injection are PQ nodes.

3. PV nodes.These nodes are also known as voltage-controlled nodes, since apartfrom Pi voltage magnitude |Vi| is kept at a specified value at eachPV node i. At node i voltage angle δi is therefore the only unknownto be solved in AC PFEs (2.10) – (2.11). The amount of reactivepower Qi is not given at this node but follows immediately from


the solution of the AC PFEs by substituting this solution in (2.11).One should think of these nodes as locations where large, control-lable power plants like fossil-fuel power stations are connected tothe grid. By tuning the turbine real power Pi is controlled, andthe voltage magnitude is controlled by adjusting the generatorexcitation. For this reason, PV nodes were also referred to asgenerator nodes before the rise of small-scale generators. However,small-scale generators like wind turbines or solar panels are ofteninsufficiently powerful to control the nodal voltage in a power grid,so corresponding nodes are often PQ nodes.Note that the AC PFEs (2.10) – (2.11) are expressed in polar form,and not for example as in (2.9). The reason is that it is the voltagemagnitude |Vi| that is given at PV nodes, and not the real andimaginary parts of Vi.

Suppose that the network consists of NPQ PQ nodes, NPV PV nodes andof course one slack node. Then we can list the numbers of specifiedquantities, available equations and state variables as given in Table 2.1.

node type # nodes Quantitiesspecified

# availableequations

# statevariablesδi, |Vi|

Slack 1 δ1, |V1| 0 0PV NPV Pi, |Vi| NPV NPVPQ NPQ Pi, Qi 2NPQ 2NPQTotal NPV + NPQ + 1 2(NPV + NPQ +

1)NPV + 2NPQ NPV + 2NPQ

Table 2.1: Summary of the AC Power Flow Equations.

There is no closed-form solution available for the nonlinear system ofAC PFEs. In fact, the solution may not exist for a given set of parameters.This case can be interpreted as the generators and the slack node beingincapable of delivering the specified demand in the power grid. In mostpractical situations a voltage collapse occurs instead. Typically, voltage


constraints (as will be described in Section 2.3) are violated long before avoltage collapse occurs, and we ignore this possibility.

Two suitable iterative methods used to solve the AC PFEs are theGauss-Seidel method and the Newton-Raphson method. The latter isknown to outperform the former in speed-accuracy ratio for all exceptvery small systems [39]. We will give an overview of the Newton-Raphsonmethod.

1. We choose initial values δ(0)i , |V|(0)i for all state variables. A typical

choice is δ(0)i = 0 for all nonslack nodes i and |V|(0)i = 1 for all PQ

nodes i. Set the index of the Newton-Raphson iteration k to zero.

2. We substitute approximations δ(k)i , |V|(k)i into the PFEs (2.10) – (2.11)

to calculate the power injection approximations P(k)i , Q(k)i for all

nodes i. Compute mismatches

∆P(k)i := P(k)i − Pi,

for all nonslack nodes i. Similarly, compute mismatches

∆Q(k)i := Q(k)i −Qi,

for all PQ nodes i.

3. We compute a modified Jacobian of the system. For notationalconvenience, we introduce the Jacobian matrix equation assuminginitially that all N nodes are PQ nodes:(

J11 J12J21 J22

)(∆δ(k)

∆|V|(k)

)=

(∆P(k)

∆Q(k)

). (2.12)

Here ∆δ(k), ∆|V|(k), ∆P(k), ∆Q(k) ∈ RN denote the vector differencesof the voltage angle, voltage magnitude, active powers and reactive


powers, respectively, at iteration k. The Jacobian block-matrices J11,J12, J21, J22 ∈ RN×N are given by

(J11)ij :=∂Pi∂δj

, (J12)ij :=∂Pi

∂|Vj|,

(J21)ij :=∂Qi∂δj

, (J22)ij :=∂Qi∂|Vj|

.

The modification of the Jacobian in (2.12) is based on the followingequivalent equation:(

J11 J12D

J21 J22D

)(∆δ(k)

D−1∆|V|(k)

)=

(∆P(k)

∆Q(k)

), (2.13)

where matrix D ∈ RN×N is diagonal with nonzero elements Dii =|Vi|(k). The matrix in (2.13) is the modified Jacobian. This modifica-tion saves the computation of half of the matrix elements as theybecome related to each other. That is, it is readily checked from(2.10) – (2.11) that

|Vj|∂Pi

∂|Vj|= −∂Qi

∂δj= |ViVjYij|cos(θij + δj − δi), for i 6= j,

|Vj|∂Qi∂|Vj|

= −∂Pi∂δj

= −|ViVjYij|sin(θij + δj − δi), for i 6= j,

|Vi|∂Qi∂|Vi|

= −∂Pi∂δi− 2|Vi|2Bii = Qi − |Vi|2Bii, (2.14)

|Vi|∂Pi

∂|Vi|= −∂Qi

∂δi+ 2|Vi|2Gii = Pi + |Vi|2Gii.

We started this Newton-Raphson step by assuming all nodes arePQ nodes. However, since not all nodes are PQ nodes certainequations and terms should be removed from (2.13) (see the nodetype descriptions at the start of Section 2.1.1). That is, since node 1is a slack node:

a) Angle δ1 = 0 and |V1| = 1 are given so ∆δ(k)1 = ∆|V1|(k) =0. Elements ∆δ(k)1 = ∆|V1|(k) as well as the first column of


J11, J12D, J21 and J22D can therefore be removed from linearsystem (2.13).

b) Mismatches ∆P(k)1 and ∆Q(k)1 are not defined so the first ele-

ments of ∆P(k) and ∆Q(k) and the first row of J11, J12D, J21 andJ22D should be removed from (2.13).

Additionally, for each PV node i:

a) The voltage magnitude |Vi| is given so ∆|Vi|(k) = 0. Elements∆|Vi| as well as the i-th column of J12D and J22D can thereforebe removed from (2.13).

b) The mismatch ∆Q(k)i is not defined so the i-th element of ∆Q(k)

and the i-th row of J21 and J22D should be removed from (2.13).

The resulting system reads(J̃11 J̃12D̄

J̃21 J̃22D̃

)(∆(δ)(k)P

∆|V|(k)Q /|V|(k)Q

)=

(∆P(k)

∆Q(k)

), (2.15)

Here J̃11, J̃12D̄, J̃21 and J̃22D̃ are the submatrices obtained by remov-ing rows and columns as described from the modified Jacobian in(2.13). (δ)P is the subvector of δ with all elements corresponding toPV and PQ nodes. |V|Q is the subvector of |V| with all elements cor-responding to PQ nodes. The division ∆|V|(k)Q /|V|

(k)Q is performed

elementwise.

4. We solve equation (2.15) for ∆(δ)(k)P and ∆|V|(k)Q /|V|

(k)Q . We compute

the next step approximations

δ(k+1)i = δ

(k)i + ∆δ

(k)i ,

for all PV and PQ nodes i, and

|Vi|(k+1) = |Vi|(k)(

1 +∆|Vi|(k)

|Vi|(k)

),

for all PQ nodes i.


5. We use these new approximations for the state variables for step2 and iterate steps 2 to 5 until ∆P(k)i , ∆Q

(k)i are within a desired

tolerance.

Once the nodal voltages are found, all connection currents Iij (and thusthe power flowing through all connections) can be computed using Ohm’slaw (2.4).

2.1.2 Fast Decoupled Power Flow

To improve the computational efficiency of the described Newton-Raphsonmethod, the Fast Decoupled Power Flow (FDPF) has been developed [81].In the last decades, the Fast Decoupled Power Flow method has becomeprevalent in industry to solve power flow equations [20, 52, 83]. Theacceleration is based on six relatively weak assumptions under which theJacobian is constant over all iterations. The resulting approximate versionof the Newton-Raphson method typically requires more iterations, buteach iteration will be computationally less intensive, and the FDPF oftenrequires less workload than the original Newton-Raphson method inSection 2.1.1. The first two assumptions are:

1. A change in the voltage magnitude leaves the flow of real powerunchanged:

∂Pi∂|Vj|

= 0,

for i, j = 1, . . . , N.

2. A change in the voltage angle δ leaves the flow of reactive power Qunchanged:

∂Qi∂δj

= 0,

for i, j = 1, . . . , N.


Then J12 = J21 = 0, so linear system (2.15) can be split into two systems:

J11∆(δ)(k)P = ∆P

(k), (2.16)

and

J22∆|V|(k)Q = ∆Q(k). (2.17)

This is the decoupling part of the algorithm. The fast part of the algorithminvolves four assumptions based on the following rules of thumb:

3. The angular differences δi − δj are usually so small that

cos(δj − δi) ≈ 1,sin(δj − δi) ≈ δj − δi.

4. The connection susceptances Bij are usually much larger than theconnection conductances Gij so that

Gij sin(δj − δi)� Bij cos(δj − δi).

5. Qi at node i satisfies

Qi � |Vi|2Bii,

6. The voltage magnitude at node i is usually close to the nominalvalue:

|Vi| ≈ 1.

We will use these approximations 3-6 to simplify J11 and J22, whoseoff-diagonal elements are given by

∂Pi∂δj

= |Vj|∂Qi∂|Vj|

= −|ViVjYij| sin(θij + δj − δi

).

= −|ViVj|[

Bij cos(δj − δi

)+ Gij sin

(δj − δi

)].


First, approximations 3 and 4 yield

∂Pi∂δj

=|Vj|∂Qi∂|Vj|

≈ −|ViVj|Bij. (2.18)

Second, approximation 5 reduces the diagonal elements (2.14) of J11 andJ22 to

∂Pi∂δi≈ |Vi|

∂Qi∂|Vi|

≈ −|Vi|2Bii. (2.19)

The resulting approximation of the first decoupled equation (2.16) reads

−

|V2||V2|B22 . . . |V2||VN |B2N

.... . .

...

|VN ||V2|BN2 . . . |VN ||VN |BNN

∆(δ)(k)P = ∆P(k), (2.20)Finally, applying assumption 6 to the first voltage magnitude of eachmatrix element simplifies (2.20) to

−BP ∆(δ)(k)P = ∆P(k)/|V|(k)P . (2.21)

Here |V|(k)P is the subvector of all elements of |V|(k) that correspondto PV or PQ nodes and the division is again performed elementwise.The approximate Jacobian −BP ∈ R(N−1)×(N−1) simply consists of theelements −Im{Yij} for all PV and PQ nodes i and j. Similarly, theapproximation of the second decoupled equation (2.16) becomes

−BQ ∆|V|(k)Q = ∆Q(k)/|V|(k)Q . (2.22)

Here the approximate Jacobian −BQ ∈ RNPQ×NPQ similarly consists of theelements −Im{Yij} for all PQ nodes i and j. Note that the approximateJacobians −BP and −BQ remain constant over all Newton-Raphsoniterations. They can be computed before iterations are commenced andeach iteration can therefore be evaluated relatively fast. One disadvantageis the fact that more iterations may be necessary due to the error ofapproximations. However, the idea is that the approximations are accurate


enough for the convergence to be faster than the convergence of theconventional Newton-Raphson method of Section 2.1.1. In the originalarticle of FDPF examples are shown where convergence required a factor5 less workload than when the exact Jacobian is used as in Section 2.1.1[81].

2.1.3 Sparse computations

The typical number N of power grid nodes depends on what is defined asone grid. Most definitions consider either a transmission grid (transport-ing electricity at higher voltage levels) or a distribution grid (deliveringelectricity to individual consumers), since grid operators are typicallyresponsible for only one of the two. Assuming this distinction, N willrange from tens to hundreds or thousands grid nodes [39]. One notableexception is the Eastern Interconnection Eastern US power grid with asmany as N = 49 000 nodes [41]. This number may increase even furtheras more power grids become connected [44, 45].

Although a power grid in theory contains M connections with N− 1 ≤M ≤ N(N − 1)/2, M is typically on the order of N. The admittancematrix Y is therefore sparse in most power grids. In this section we willexplain how to benefit computationally from the sparsity of Y. To com-pute the mismatches ∆Pi/|Vi| as proposed in the previous subsection, oneneeds to evaluate the current Newton-Raphson iteration approximationfor

Pi/|Vi| =N

∑j=1|YijVn| cos(θij + δj − δi),

=N

∑j=1|Vj|(

Gij cos(δj − δi) + Bij sin(δj − δi))

,

for all nodes i. The second equality follows from a trigonometric identityand the definition (2.8) of Y. One can write this in the vector form

P/|V| = A(G, B, δ)|V |, (2.23)


Figure 2.2: Sparsity of Y for the IEEE-30 and IEEE-118 test cases, respectively.Test cases can be found in [87]

with vectors P, |V |, δ ∈ RN , where the division on the left-hand side isperformed elementwise, and where matrix A(G, B, δ) ∈ RN×N dependson G = (Gij), B = (Bij) and δ:

A = (Aij), with Aij = Gij cos(δj − δi) + Bij sin(δj − δi).

Now note that A will be at least as sparse as Y = G + ıB. Therefore,to evaluate (2.23), workload will be reduced by only computing thenecessary terms in the summand by precaching the indices of nonzeroelements of Y. Neglecting the cost of inversion of matrices BP and BQ —which is reasonable in Monte Carlo simulations of the following chapterssince we can reuse the inverse every time step and sample — it is readilychecked that the computational complexity of one sparse FDPF iterationgrows as O(M), with M the number of power grid connections. Thiscompares to O(N2) for nonsparse FDPF as described in Section 2.1.2.If used in a Monte Carlo simulation, conventional Newton-Raphson asexplained in Section 2.1.1 will certainly be computationally inferior toFDPF: at every time step in every sample a linear system must be solvedinstead of a matrix-vector multiplication. An experiment comparingsparse FDPF and the conventional Newton-Raphson method showed


a decrease in CPU time for all but the smallest IEEE test cases (resultsnot shown here). In the IEEE-300 test case with N = 300, M = 411 themethod converged around twice as fast, confirming that a sparse FDPFmethod accelerates the conventional Newton-Raphson method for theAC PFEs. We will use both sparse computations and the FDPF methodto solve the AC PFEs in this work.

Using Table 2.2 we will give an insight in the computational costs ofdifferent parts of the sparse FDPF solver for different IEEE test cases [87].All average CPU times are based on 100 measurements. The initialization

IEEE-N test case, N = 14 30 57 118 300

Initialization 0.112 0.15 0.26 0.71 6.62

Inversion of B 0.084 0.13 0.23 1.09 9.66

Inversion of B̄ 0.031 0.061 0.13 0.25 4.81

Sparse indexing 0.031 0.037 0.060 0.17 1.33

Newton-Raphson 0.52 0.46 1.16 1.16 6.88

Post-processing 0.07 0.11 0.26 0.61 6.53

Total 0.85 0.95 2.09 4.00 35.8

# Newton-Raphson iterations 7 9 16 7 16

Table 2.2: Average CPU times (ms) of parts of the sparse FDPF solver.

refers to the extraction and processing of input data. The first inversion isthat of the Jacobian in (2.21). The second inversion is that of the Jacobianin (2.22), which is a smaller matrix since the PV node equations areomitted here. The part of sparse indexing refers to the collection ofindices of nonzero elements of Y, as well as the corresponding indices ofother matrices. In the Newton-Raphson loop, the solution for the statevariables |V | and δ is derived. From this solution, all power injections,connection currents and connection power flows are derived in the post-processing part. One can see from this table that for larger networks,the CPU time of the inversions becomes significant. For N = 118 orN = 300 this part is computationally more intensive than the part of theNewton-Raphson iterations. Nevertheless, Monte Carlo simulation will


require both inversions only once after which the resulting inverse canbe used each time step and sample path. Therefore, the workload of thetwo inversions will in our case most probably be insignificant, and wewill not attempt to improve it.

2.2 the direct current power flow model

The alternative Direct Current (DC) PFEs can easily be derived from theFDPF method described in Section 2.1.2. The two main assumptions arethe following:

1. The voltage magnitudes are assumed to be equal to the nominalvalue: |Vi| = 1 at each PQ node i. Note that the FDPF methodassumed this to be true for some values |Vi| in the Jacobian only,whereas the DC model assumes nominal voltages in the systemitself.

2. Shunts yii are ignored in the admittance matrix Y (see (2.7)). Theresulting susceptance matrix is denoted by B′P.

The first assumption implies ∆|V|(k)Q = 0 in (2.22) and thus this equationbecomes redundant. It remains to solve (2.21) for the voltage angles δi atall nonslack nodes only. Since |Vi| = 1, this linear system becomes

−B′P

∆δ2

...

∆δN

=

∆P2...

∆PN

.The computational complexity of the algorithm is O(M) when usingsparse computations. This is the same as that of one sparse FDPF iterationof the AC PFEs and DC solvers are therefore faster than AC solvers.Furthermore, the linear form of the DC model enables a closed-formsolution for the state variables. For this reason an analytic approach ismore often viable when assuming the DC power flow model than whenassuming the nonlinear AC power flow model. However, the DC powerflow model assumes voltages to be equal to the nominal voltage value.

2.3 grid stability 29

This is a strong assumption, so the DC model is considered less accuratethan the AC model for power grids with alternating current.

2.3 grid stability

We call a power grid stable (also called in normal operation) if the followingconstraints (also called operating limits) are satisfied [96, 7]:

1. Connection constraints.For each connection (i, j), the temperature Tij(t) should be boundedat all time t:

Tij(t) < Tmaxij . (2.24)

Violation of this stability constraint will cause the correspondingline to loose its tensile strength or sag. In turn, this will influencethe admittance of the line, although we neglect this phenomenon inthis thesis. Grid operators have to take this constraint into accountwhen dimensioning a new cable or line. One sufficient condition forconstraint (2.24) to hold is that the connection current is bounded

|Iij(t)| ≤ Imaxij , (2.25)

or equivalently that the power flowing through the connection isbounded:

|Pij(t)| ≤ Pmaxij . (2.26)

This condition is in general too strong as the temperature incurssome lag time, as we will illustrate in Chapter 4.

2. Voltage (magnitude) constraints. The voltage magnitudes should liebetween acceptable bounds at all PQ nodes at all time t.

Vmin ≤ V(t) ≤ Vmax, (2.27)

If a voltage constraint is violated, equipment connected at thecorresponding node will get damaged.


3. Reactive power constraints. The reactive power should lie betweenacceptable bounds at all PV nodes.

Qmin ≤ Q(t) ≤ Qmax. (2.28)

Grid operators are responsible for monitoring the stability of the powergrid and act on predicted violations of stability constraints (2.24), (2.27)and (2.28). Conventionally, corrective actions like rescheduling generationhas been used as a first attempt to avoid predicted violations. If not allviolations can be prevented in this way, the grid operator will curtail loads.That is, at specific nodes demanded power is not delivered. However, asexplained in Section 1.1, grid operators can not easily reschedule genera-tion in privatized electricity markets since power suppliers are marketplayers just as power consumers. Therefore, rescheduling generation canbe viewed as a curtailment that is similar to a load curtailment, andwe regard both as a power curtailment in general. In fact, to resolve gridinstability, an Optimal Power Flow problem has to be solved, where themost economic dispatch of both generation and consumption is chosensuch that all constraints are satisfied.

Optimal Power Flow is a research area in itself (see e.g. Solimanand Mantawy [80, Chapter 5]), and is outside the scope of this thesis.Instead, we assume that a violation of a stability constraint like (2.24),(2.27) or (2.28) immediately induces a power curtailment and is assuch undesirable. However, the complexity of many Optimal PowerFlow problems justifies the aim of this research to develop acceleratedsimulation techniques of grid violations occurrences: a natural extensionof this research would be an Optimal Power Flow solver where such asimulation method evaluates each state in the optimization procedure. Inthis way relatively complex Optimal Power Flow models incorporatingpower injection uncertainty can be solved within a reasonable amount oftime.

2.4 deterministic heuristics assessing power system reliability 31

2.4 deterministic heuristics assessing power system reli-ability

In this chapter we introduced deterministic models for power flowanalysis. Many deterministic power flow analyses were developed inthe twentieth century when centralized, controllable (fossil-fuel) powerstations supplied the electricity in a ‘top-down’ fashion. DeterministicAC power flow equations were used to compute the nodal voltages frompredicted constant power injections (or piecewise constant functions oftime). Using these values the grid stability could then be evaluated.

Instead of using a probabilistic approach, the grid state can in principlebe evaluated using different scenarios, including the scenario undernormal operation and specified worst case scenarios. A widely usedexample of a worst case scenario is the n − k criterion [48]: given ngrid components (connections, generators, transformers, etc.), is thegrid stable if k components — with k = 0, 1, 2 being typical values— fail? Iterating over all possible combinations of component failuresgives an insight what types of contingent events the power grid canwithstand. However, the probability that a combination of componentsfail simultaneously is not taken into account. Therefore, some scenariosmay be very unlikely, or even worse, a likely and catastrophic scenario ofmore than k component failures is neglected in the analysis. Furthermore,only the state of components are assumed uncertain, and not the powerinjections.

Other deterministic approaches have been used to account for powerflow uncertainty. One example is the Strand-Axelson model [82] thatheuristically relates the maximum load Pmax to the annual energy con-sumption Ey by a consumer:

Pmax ≈ αEy + β√

Ey. (2.29)

The coefficients α and β have to be determined empirically, and willtypically depend on the considered area and the connection type [98].Note that the maximum loads of different consumers Pmaxi will in generaloccur at different times. This implies that the maximum load Pmaxcable of a


Supply 2

1

3

Pmax1

Pmax2

Pmax3

Pmaxcable

Figure 2.3: As peak loads of consumers do not occur simultaneously in general,Pmaxcable = ∑

3i=1 P

maxi does not necessarily hold.

cable will be less than the sum ∑ni=1 Pmaxi over all lower level lines fed by

this cable (see Figure 2.3).The Rusck model [74] heuristically relates Pmaxcable to P

maxi assuming

homogeneous patterns of all consumers i:

Pmaxcable ≈ nPmax,1(

s + (1− s)/√

n)

. (2.30)

Here n is the number of consumers and simultaneity factor s is to befound empirically.

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Assessing power grid reliability using rare event simulation · Any power ﬂow model starts with the deﬁnition of its topology. The topology can be modeled by a connected graph